Optimization of a Semiflexible Demand-Responsive Feeder System in Suburban Areas Using a Memetic Algorithm

. Traditional bus operations in suburban areas are inefcient due to their fxed routes and timetables. Since suburban operations deal with low demand spread in a large area, the service stays underused for most of-peak hours. In order to render the operation proftable and increase the number of passengers on each bus, operators reduce the frequency of the service, which results in an increase of passenger waiting time for the service. As a solution to this problem, this paper introduces a demand-responsive public bus system that aims to adjust routes and timetables of a semifexible system to the demand for transportation. Te operation still ofers a reliable service like the traditional system but aims to reduce the passenger travel time. A memetic algorithm is developed to optimize this demand-responsive system. For a network with 25 bus stops served hourly by three lines and with an average demand of 20 requests per hour, the memetic algorithm is demonstrated to reduce the passenger waiting time with almost 50% in comparison with a traditional system operating in the same network with fxed routes and timetable.


Introduction
Te growth of (sub)urban areas around big cities, such as new neighborhoods or settlements that operate as commuting towns, puts pressure on public transportation systems.Tese new urban areas generally have a low population density spread over a large area.Direct connections or extensions of the current routes of the public transport system potentially lead to a nonproftable service.In order to improve the public transportation system in these areas, the most applied solution is the implementation of feeder lines.Feeder lines zigzag through the suburban area to bring passengers to a hub station where a high-frequency service ofers a direct and fast connection to the demand centers (city center, industrial zones, and schools).
Feeder lines partially solve the efciency problem in suburban areas with low demand, especially during peak hours.However, when demand is lower during of-peak hours, feeder lines with fxed routes and timetables are no longer efcient.Teir service is typically optimized for the demand predicted based on historical data, without considering external factors that might infuence the daily demand using the service, such as weather conditions or nonregular events.
Another approach to address this type of demand are fully fexible demand-responsive (taxi) services, which operate without using fxed routes or timetables but pick up and drop of passengers according to their needs in a service complimentary to the traditional transit system, typically in a door-to-door service.In the OR literature, such systems are often modeled as a variant of the Dial-a-Ride Problem (DARP).Despite being a very responsive service, the operation of such systems in suburban areas with low demand can also lead to low seat occupancy, and consequently, an expensive operation.
Tis paper aims to investigate the applicability of a semifexible demand-responsive transportation system replacing a traditional feeder system in suburban areas.Te fact that this system is semifexible is a key novelty.Tis means, on the one hand, it has some characteristics of traditional systems, i.e., a regular frequency of operation, predefned standard routes, and a stop-based service.Tis "backbone" makes this system recognizable for the users and lowers the threshold for using it.Tus, this system still serves the demand by relying on the structure of the traditional feeder system.On the other hand, it is a demand-responsive system, allowing modifcations to the standard routes and adjusting the timetable of each trip while respecting the initial frequency.Terefore, our proposed system falls in the category of a semifexible demand-responsive feeder systems (DRFS) and intends to ofer an optimized service with some level of responsiveness to the actual demand for transportation yet still based on a service running regularly.It should also be noted that the same resources are being used as in the traditional system, so the operator costs remain the same and the focus will be on ofering a better service, with roughly the same operational costs.
As the traditional system, our semifexible DRFS is stopbased.It means that passengers willing to submit a request must indicate beforehand their origin bus stop and walk to the stop at the preferred departure time.Requests for a trip can be submitted until just before the operation starts.Based on a list with all passenger requests, the routes and departure times for the buses operating each line are optimized for each hour of operation of the services.Te objective when optimizing the operations of this DRFS is to minimize the sum of the travel times for all passengers, from the moment they arrive at the bus stop until they arrive at the hub station.Terefore, a memetic algorithm (MA) is introduced to optimize the performance of the DRFS from the users' perspective.
Te main contributions of this paper are that we introduce and analyze a new, semifexible, demand-responsive feeder system, designed for low demand suburban areas.Diferent from the other DRT systems, our semifexible system tries to combine the benefts of a fxed system and a fully fexible DRT.Furthermore, a memetic algorithm is designed for optimizing the performance of this new system when serving a set of registered requests.After evaluating the performance of the algorithm, the performance of the system is analyzed and compared to the performance of a traditional feeder system.Tis allows determining under which circumstances this new system should be considered in practice.
Te related literature on public bus transportation systems is discussed in the next section, both for feeder systems in general and for the DRFS.Te assumptions for the DRFS proposed in this paper are described in Section 3. Te memetic algorithm is discussed in detail in Section 4. Section 5 presents the results of the experiments.First, the efciency of the memetic algorithm is evaluated by comparing its results with an optimal solution obtained through a mathematical model.Ten, the DRFS is compared to the TFS.Several instances are generated, varying the list of requests in each instance.Five diferent experiments are performed for these instances.One last experiment considers a large network and instances with more requests.Section 6 concludes the paper and discusses opportunities for further research.

Literature Review
Tis section frst discusses the uprise of demand-responsive systems, with diferent levels of fexibility.Ten, the DRFS is positioned in the state of the art and various closely related systems are discussed.
As an alternative for traditional systems, demandresponsive systems emerged in the last decades and their popularity has been rising over the years.One of the initial, yet still popular, demand-responsive systems is a door-to-door dial-a-ride (DAR) system with the purpose of transporting reduced-mobility or elderly passengers, typically running in parallel to a fxed service [1].Such a system is also used as a transportation mode for rural communities without scheduled bus services.Tis fully fexible system is typically operated based on reservations.After the surge of more innovative solutions and intelligent transportation systems (ITS), more demand-responsive transport systems were developed for a wider variety of purposes, complementing the fxed transportation systems [2][3][4][5].
Several papers explore case studies to improve the current transit operation systems in rural areas (i.e., see [6]), the implementation of alternative DRT systems and other modes of operation (i.e., see [7]), or the optimal redesign of the DRT systems to increase efciency [8].Te common characteristic identifed in transit operation in suburban or rural areas is the necessity of subsidies to make the service viable.Terefore, it should be noted that our semifexible DRFS uses roughly the same resources as the traditional feeder systems.
Depending on the transportation system proposed, diferent levels of fexibility are considered that may afect stops, routes, timetables, or prebooking requirements [9].On the most limited fexibility level, systems have been proposed with fxed bus stops, routes, and timetables, but which run only when a booking is made.On the other end of the spectrum, a fully fexible system operates without predetermined bus stops, routes, or timetables, serving users in a service resembling that of a taxi.Tese fully fexible systems typically run using smaller vehicles, and routes are created according to passengers' requests in a ridesharing format, with fxed or variable fares according to the travelled distance and service availability.Some papers compare these fully fexible systems with other systems or traditional bus systems using simulations [10][11][12].Other papers focus on developing guidelines for the operation and classifcation of these systems [13].
According to a recent and comprehensive survey and classifcation of demand-responsive public bus systems [14], our DRFS corresponds to a many-to-one, semifexible, stopbased, static demand responsive bus system.Tis means it is feeder line ("many-to-one"), with fxed stops ("stop-based", not "door-to-door"), but fexible routes and timetables ("semi-fexible"), which are designed before the operation starts ("static").Moreover, with the objective of minimizing the passenger travel time, it takes the passengers' perspective.Only a few of these systems have been considered before [15,16], but they consider systems with a completely different design.
Still, other semifexible feeder services, combining characteristics of both the traditional and demand-responsive systems, can be found in the literature.For example, a customized bus (CB) ofers a regular operation optimized to passengers' requests, determining routes according to the pick-up and drop-of locations of passengers.Te operation is adjusted on a daily basis, based on the clustering of demand in the service area [17][18][19].A demand-responsive connector (DRC) operates a fully fexible door-to-hub service in cycles [20].However, the cost of such taxi-like or shuttle transit mode requires large cities with dense populations for a proftable operation [21].
Te resulting system from merging a demand-responsive and a traditional system can generate a substitute for the traditional system or a complementary system running in parallel.Nourbakhsh and Ouyang [22] explore a fexible service for low-demand areas, ofering a door-to-door service that substitutes the traditional operation.Tis system, however, is applied only for a grid network.In another complementary system for high-demand areas with a grid structure, Uchimura et al. [23] design a hierarchical structure.On the upper level, traditional mass transit lines connect diferent areas with an express service.Te intermediate layer operates an intercommunity traditional transit system ofering hub stations where passengers can switch to the last layer, which is the intracommunity service designed as a fully fexible door-to-door demand-responsive system.Only a few of these systems could be used as a replacement system in low demand areas.
Another aspect taken in account for semifexible services is the way to structure the routes for the buses.Pratelli et al. [24] design a demand-responsive feeder system were part of the stops are fxed, as in the traditional system, but the demand-responsive operation allows deviations to include a limited number of extra requests from optional stops.A methodology to choose the most suitable policy is defned according to the level of demand and the accommodation rate [15].In our semifexible DRFS, the structure of the lines from the traditional feeder system are kept, such as a fxed terminal to begin the service and a common destination for all requests.Limited fexibility from this original line allows optimizing the operation in of-peak hours considering the preferred departure time of the passengers.
To conclude, our DRFS operates in suburban areas as a many-to-one, semifexible, stop-based, static demand responsive bus system, in order to substitute a traditional system in of-peak hours.Terefore, it keeps the traditional feeder systems' structure but includes fexibility in order to allow passengers to request a trip to the hub station at their preferred departure time.Mainly, the aspects that it is a semifexible system and that it operates in of-peak hours making it diferent compared to other systems in the state of the art.

Problem Statement
Tis section discusses the assumptions on the semifexible demand-responsive feeder system (DRFS) and the required input data.Te starting point is a traditional feeder system (TFS) operating a fxed number of bus lines with standard routes and a regular timetable in a suburban area.Tis system is optimized for the average daily demand.
Te DRFS is designed to replace the TFS during of-peak hours and will modify the standard route for each bus line towards the hub station and the departure time of the scheduled buses based on passenger requests.For every operation period of the problem, a list of passenger requests is available.Each request represents a trip by a passenger, with a desired bus stop and preferred departure time to start the trip.Te standard route of each bus line can be modifed with one or more shortcuts and limited detours.With "limited detours," we mean that in each detour at most two additional stops can be visited before returning back to the standard route.Tis assumption guarantees, on the one hand, that lines are still similar to the standard lines, and on the other hand, that sufcient opportunities are available for shortcuts to skip bus stops without passengers and making detours when appropriate.For the purpose of limiting the fexibility in this study, the value was set to at most two additional stops per detour, creating a balance between a fully fexible DRFS and the TFS with fxed lines.Tis value, however, could be infuenced by several other factors in reality, such as the availability of an alternative route between stops, or the ability to visit multiple diferent stops with only a small detour.Tis results in a limited pool of possible routes for each bus line.Figure 1 presents an example of the DRFS operation.As in the traditional system, a bus line is indicated by a unique nomenclature (a number, a code, or a color, for instance) and this nomenclature is used by passengers to identify the bus line they need to their destination.In Figure 1, the standard line of the TFS is indicated as a solid line, with bus stops 6, 5, 4, 1, and the last stop S. In the DRFS, it can be decided, before the operation starts, to include the dashed detour, picking up passengers at stops 3 and 2 and skipping stops 5 and 4. Tis result in two possible routes for this line: the standard line and the line with the dashed detour.

Assumptions for the DRFS.
Te DRFS runs in a suburban area represented by a graph G � V, E { }, where V is the set of bus stops (i � 0, 1, . . ., n), and i � 0 representing the hub station.Edges E indicate links between bus stops i ∈ V and j ∈ V at a travel cost of c i,j .Some initial assumptions for the operation are as follows: (i) Te objective of the DRFS is to minimize the total travel time of the passengers; (ii) Te service operates for n consecutive periods; (iii) Tere are a fxed number of lines connecting all the bus stops to the hub station; (iv) All lines start at a predefned terminal bus stop and head to the hub station; Journal of Advanced Transportation (v) Each bus is associated with a line, following a route from the pool of routes generated for that line; (vi) Te pool of routes includes all feasible routes for each line, considering stop skipping and limited detours; (vii) Te route length is limited to the duration of the operation period (p op ); (viii) Te buses depart from the terminal and arrive at the hub station within the operation period; (ix) Te departure times of the buses are not fxed but can be adjusted within the operation period; (x) Te bus capacity is assumed to be sufcient to address all requests.
For each period of operation, there is a list of requests representing passengers that want to use the service during that period.Each request is composed of an origin bus stop and a preferred departure time (p r ) representing the moment a passenger will be ready to take the bus towards the hub station.A request can be accepted or rejected.If accepted, it means that this passenger will be picked up by one of the buses in that operation period at pick-up time (p p ) and will reach the station at drop-of time (p d ).If a request is rejected, it means no bus will pass by the passenger's bus stop after its preferred departure time, and a penalty will be associated with this request.Rejected requests are then included in the request list for the following operation period.After the last optimized of-peak period, it is assumed that again the TFS is operated, serving all stops.Terefore, all passengers will be served.
Te main objective of the DRFS is to minimize the passenger total travel time (p tt ) by selecting, in each period, the best route and best departure time for each bus.Te total travel time consists of the passenger waiting time (p wt ) at the bus stop and the passenger in-vehicle time (p vt ) until passengers arrive at the hub station.Terefore, the total travel time for each passenger given in the following equation is the diference between drop-of at the hub station and the preferred departure time: In order to determine the penalty for a rejected request, it is assumed that passengers will be waiting for a bus until the next operation period.Since the routes for the next period are not defned yet, and since all buses depart from the terminal and arrive at the station within the operation period, it means that, in the worst case, rejected requests will arrive at the station by the end of the following operation period.Terefore, a passenger penalization (p pen ) is given by the time diference between the passengers' preferred departure time and the end of the following operation period given by the following equation: (2)

Generation of the Pool of Routes for the Lines.
Te problem formulation of the DRFS uses a limited pool of alternative routes for each bus.In this subsection, we develop a method to generate such a pool.Te proposed semifexible DRFS focuses on operating a fexible service but keeping the main characteristics of the TFS.Terefore, we start from the standard (TFS) route of each line and then generate a set of alternative routes using "stop skipping" and "limited detours."Te standard routes are optimally designed for the TFS, serving all bus stops and efciently accommodating all the demand over multiple periods.In the DRFS, buses do not follow these standard routes, but decide for the best route according to the requests for each operation period.Stop skipping and limited detours are considered.Stop skipping means that it is possible to skip one or a sequence of bus stops in the standard route.Detours mean that it is possible to include additional bus stops in the standard route.Since the DRFS is not a fully fexible system where buses are allowed to take any possible route, we limit the detours to including at most two other bus stops before returning to a bus stop in the standard line.However, it is possible to combine detours with stop skipping.Furthermore, cycles are not allowed, which means that a bus stop cannot be visited twice by the same bus.Moreover, a limit is set on the maximum route length to control the maximum travel time a bus can ride.It should be noted that overlaps between diferent lines are possible, resulting in more than one line serving the same bus stop.
By considering the limited detours, it is possible to precalculate all possible routes for each line in the DRFS.Tese alternative routes are then stored in a pool of routes that are used by the memetic algorithm instead of recalculating all possible movements while exploring solutions for the routes.As an example, Figures 2 and 3 illustrate a TFS operation in a grid network composed of nine bus stops and a station S, served by a red line (Figure 2) and a blue line (Figure 3).Figures 2 and 3 also indicate the possible detours or stop skipping for these lines, based on the previously  Journal of Advanced Transportation stated assumptions.For instance, the original route of the blue line includes the sequence of stops 9-8-7-4-1-0.When the bus arrive at the stop number 8, it is possible to include a detour to stop number 5 instead of continuing the initial route to stop 7. From this point, the bus can return to the original line, visiting stop number 4 and skipping the bus stop number 7, or include another detour to visit stop number 2. After this second detour, the bus is obliged to return to the original line.Te stop 3 cannot be included in the blue route since it would require a detour of more than two stops to return to the standard route.Tat is, from stop 9, it would require a detour to stop number 6 and then to stop number 3, but then it would require at least one more stop to return to the original line, which is not allowed.

List of Requests and Operation
Period.Every passenger that wishes to use the system submits a request composed of a bus stop and a preferred departure time.It means passengers are planning to arrive at the given bus stop and will be waiting for the bus from that moment on.Since the ofpeak operation is divided in periods of operation, requests have to be submitted before the start of the corresponding operation period.Once the buses of an operation period start running, no new requests for that operation period are allowed.Terefore, the requests considered for the optimization of an operation period are those related to that operation period and unserved requests from previous operation periods.

Memetic Algorithm Approach
Optimizing the performance of the DRFS discussed in the previous section is addressed with a memetic algorithm (MA), combining an evolutionary heuristic suitable for exploring large areas of the search space in coordination with repair and improvement functions, acting as local search techniques, to improve solutions to ft the preferred departure time of passengers.Te framework of the proposed MA uses diferent iterations to confgure the buses' operation, i.e., route assignment and departure time for each bus.An overview of the MA is presented in Algorithm 1. Figure 4 illustrates the fowchart of the MA.For each period of operation, the requests received for that period, the rejected requests from previous periods and the list of possible routes for each line are considered as input.Subsequently, a generation with initial solutions is created and the ftness value for each solution is calculated.Ten, the process of selection, crossover, mutation, repair, and improvement is repeated until a given number of generations is produced without further improvement of the best solution.
For each operation period in the operation horizon, a solution is represented by a list of bus stops and a departure time from the terminal for each bus line of the service.Te route is represented by a sequence of bus stops and the departure time with a delay from the beginning of the operation period.For instance, a possible representation for the red line service on Figure 2 could be the route that departs from the terminal 20 minutes after the operation period begins and takes a shortcut straight to bus stop 2, where the standard route is followed until the station.

Population Generation.
Te MA approach aims to identify, for each operation period, the solution that minimizes the total passenger travel time.Te frst step of the MA is to populate the initial generation with solutions.Since all buses must arrive at the station before the end of the operation period, the route length determines the maximum delay time a bus can have.Terefore, the route is assigned frst; selecting a random route from the pool of routes for each line, and then, a departure time is assigned randomly limited to the maximum delay time.Tis random generation tries to cover a large number of possible combinations of routes and departure times as possible, which are then subjected to further improvements in the following generations.

Evaluation of Solutions.
After assigning a route and a departure time to each line of the operation period, it is  possible to generate the timetable for all bus stops and the arrival time of the buses at the station.Tis information is used to evaluate the objective function value of the solution using the list of requests for the current operation period.Te ftness of a solution is given by the sum of passenger travel time for all requests in the list of requests (r ∈ R) as follows: Objective function value �  r∈R p tt + p pen . ( For every request in the list of requests, the passengers' preferred departure time (p r ) is compared with the timetable for that bus stop.If a passenger arrives at the bus stop before the bus, the request is accepted and p p and p d are updated.If the passenger arrives at the bus stop after the bus passed by, or no bus visits the stop in that operation period, the request is penalized, and this request is included in the list of requests for the next operation period.If more than one bus from diferent lines can serve a passenger in the same operation period, the one that arrives earlier at the station is chosen.It should be noted that this is not necessarily the frst bus serving the stop.
After evaluating the new solutions in a generation, they are sorted according to the ftness value, and the solution with the smallest ftness is set as the best solution of the generation.Also, the top 5% of the solutions are set as good solutions for the elitist crossover method (below).If the population size is smaller than 100 solutions, at least fve solutions are set as good solutions.Te evaluation ftness function is presented in Algorithm 2.

Crossover Operation.
Te frst operation to populate new generations is the crossover.Te MA uses two diferent methods: an elitist and a random crossover, with a probability of 50% each.In the elitist method, at least one of the

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Journal of Advanced Transportation parents is selected from the group of good solutions.Te other parent is randomly selected.In the random crossover, both parents are selected randomly from the general population.Once the parents are selected, the crossover selects the routes of the lines and the departure time from the parents to generate two new solutions.It is done by randomly assigning to each line of one new solution, the route and departure time used by a randomly selected parent on this line.Te nonselected route (and departure time) for that line is then assigned to the second new solution.Te crossover operation is presented in Algorithm 3.

Mutation Operation.
Te mutation operation includes variability in the generations by changing routes and departure times of a solution after the crossover method, with a 5% probability of occurring.A mutation can be a small or a large mutation, each with a 50% probability.In the small mutation, one route and departure time of the solution is randomly modifed.In the large mutation, all routes and departure times are generated again for that solution, resulting in a completely new solution.Tese new solutions are introduced to stimulate diversifcation in the population.After the mutation operation, the new solution is included in the new generation list, whether mutated or not.Te mutation operation is presented in Algorithm 4.

Repair Operation.
After populating the new generation, all solutions pass through both the repair and the improvement operations.Te repair operation modifes the solution, trying to avoid penalized requests and accepting them in the current period.First, all rejected requests in the list of requests are categorized according to the type of penalty, i.e., whether the penalty results from an early bus service or from an unserved bus stop.An early bus service means a passenger arrives at the requested bus stop after the last bus has already served that bus stop, and consequently, this passenger is not served.In the case of an unserved bus stop, the bus stop is not visited by any bus in the current solution, and passengers requesting travel from these stops are forced to wait for the next period.Once all penalized requests are categorized, one of them is randomly selected to be "repaired" by forcing this request to be served on one of the lines.If the request is rejected due to early service, the repair function tries to delay the corresponding bus as much as possible, so the bus arrives at the bus stop at the same time as the passenger.It consequently delays all arrival times for all the bus stops in the route and thus increases the waiting times for other passengers.Obviously, this delay is limited by the fact that the bus should arrive at the hub station before the end of the operation period.
A more complex procedure is executed if the request is rejected due to an unserved bus stop.First, one of the possible lines that may serve the bus stop is selected.In the example of Figures 2 and 3, for instance, bus stop three can only be served by the red line, bus stop seven only by the blue line, and all the others bus stops by both lines.Second, considering the current route for the selected line, all bus stops where passengers are being served by this route are marked, and a new route including all these marked bus stops and the bus stop of the rejected request is chosen from the pool of routes for this line.Finally, the procedure checks whether the departure time needs to be adjusted, either because the new route is longer than the previous or because the arrival time at the included bus stop is still not serving the penalized request.
An example of this procedure is illustrated in Figure 5.In this example, there is a penalized request at bus stop four, not served by neither the red line (a) nor the blue line (b), since this bus stop can be included in both the lines, stops with served passengers on the current route of the lines are marked: stops 8 and 1, if the red line is chosen, and stops 8 and 2 for the blue line.Tere are no possible routes including the stops 8, 4, and 1 in the same route for the red line, therefore, this option is discarded.For the blue line, it is possible to change the route to the sequence illustrated on Figure 5(c).Te last check adjusts the departure time for the optimized request.Te bus in the blue line departs at the beginning of the operation period.It will serve bus stop 4, 15 minutes later with the new route.If the passenger arrives any time earlier, there will be no adjustment in the departure time of the bus.If the passenger arrives later, the departure time of the bus will be delayed as much as possible, so this passenger does not have to wait for the bus, respecting the maximum arrival time at the station.After fnishing the repair operation, the new solution is re-evaluated, and if the ftness is improved compared to the previous one, the new repaired solution replaces the earlier solution in the new generation.Otherwise, it is discarded.4.6.Improvement Operation.Te last operation included in the MA is the improvement operation.While the repair operation tries to delay the departure time of buses or to include stops not served in the routes, the improvement operation advances the departure time of a bus or makes the route more direct to the destination.Te operation starts with the selection of a random request from the list of accepted requests.After checking which bus is currently serving the request, two steps are executed.First, it tries to fnd a new route aiming to reduce the in-vehicle time for the passenger of this request.As in the previous operation, all bus stops serving passengers along the current route are marked.Ten, a new route is selected for this line only when there is a shorter route for the selected request with all the marked bus stops along the route.Second, the bus departure is set earlier as much as possible to reduce the waiting time for this request.Te new solution replaces the earlier one in the new generation if the improvement operation improves ftness.
Te operation of the improvement function is illustrated in Figure 6.In this example, there is a request for bus stop 2 with the preferred departure time 30 minutes after the period starts, and this passenger is served with a bus from the blue line following route illustrated on the left in Figure 6.Te current departure time from the terminal is 25 minutes after the beginning of the period, and the bus is picking up the selected passenger at time 40.Terefore, the waiting time for this passenger is 10 minutes.In the improved solution, on the right in Figure 6, the route is kept unchanged since there is no route in the pool of routes with a shorter invehicle time from bus stop 2 to the station.Ten, the bus is set to depart 10 minutes earlier, consequently arriving earlier at every bus stop along the way and picking up the passenger at bus stop 2 without any waiting time.If other passengers are being served by this bus, there are two options: their waiting times are reduced by 10 minutes, or their requests are rejected in this solution.After updating the route and the timetable, the new solution is re-evaluated, and if the ftness is better, the new solution replaces the earlier one in the new generation.Otherwise, it is discarded.

Results
In this section, results for the MA are presented in six analyses: First, benchmark instances for the DRFS are created and the parameters of the MA are evaluated in a sensitivity analysis.Te MA is then validated through a comparison with a mathematical model that solves a very similar but somewhat easier problem to optimality.Second, the MA solutions for the benchmark instances are compared with those of a TFS.Tird, instances are generated with variations in the number of lines.Fourth, more complex instances without restriction on the possible detours are created.In these instances, buses are allowed to take any route in the network from terminal to station.Finally, the MA is evaluated on larger instances, with 70 bus stops in the area instead of 25 bus stops.Tese two last experiments allow evaluating the viability of the MA to fnd local optima for complex instances and to check the performance of the DRFS.
5.1.Network and Instances.Since this DRFS was not studied before, there are no benchmark instances available to evaluate the performance of the MA.Terefore, new instances are created based on a graph corresponding to a suburban area served with a feeder bus system.It is composed of 25 bus stops connected to a transfer station (S) and served by three lines.All lines start at bus stop/terminal 25.Each bus stop expects 1 to 6 passengers, with a total of 100 requests over fve consecutive operation periods.Te time a bus takes from one bus stop to another is equivalent to the Euclidian distance between the bus stops, converted to minutes, and presented in Figure 7. Te length for the standard lines for the TFS and the number of alternative routes in the pool of routes for each line in the DRFS are presented in Table 1.Te lines for the TFS are optimized to serve the total demand presented in Table 2. Tese lines are obtained with an exact optimization approach developed for the bus line planning problem considering the average daily demand expected at the diferent bus stops [25].Te objective is to minimize the passenger travel time from the diferent bus stops to the hub station.It results in the optimal standard routes for the lines of the TFS.In a second step, the frequency of the service is set to one bus per line per operation period.Tis procedure is similar to the planning of traditional feeder systems in practice.
One instance with fve operation periods is shown in Table 3.In each operation period of this example, requests are concentrated in 9 to 15 diferent bus stops of the 25, corresponding to, on average, half of the bus stops per operation period.Tis means that in the total operation horizon, there are 50% of stops without requests (50% of empty bus stops).Several instances with random distributions of demand were created, categorized according to the number of empty bus stops in the total operation: 30%, 50%, or 70%.

Sensitivity
Analyses for MA's Parameters.Tis sensitivity analysis examines the parameters for the MA, i.e., the size of the population in each generation and the stop criteria in the search for improvements.In order to somewhat limit the computation time required to optimize an operation period, population sizes of 50, 100, or 200 are considered combined with a maximum number of generations without improvements of 20, 50, or 100.Beyond these parameters, the total number of generations was limited to 1.000.During the experiments reported in this paper, however, this value was never reached.
Te results for this sensitivity analysis are presented in Table 4 for 15 instances grouped in three categories according to the percentage of empty stops (30%, 50%, and 70%), with fve instances for each group.Te results present the average passenger travel time in minutes, and the computation time in seconds.Considering the limited available time for computations just before each operation period, and considering the quality of the results, we conclude that a population size of 100 solutions and a stop criteria of 50 generations without improvements are appropriate parameter values, increasing the population size from 100 to 200 or the maximum number of generations without improvement from 50 to 100 produced the same passenger travel time on average (31.8 min) for these 15 instances, but the computation time increased from 55 to 126 and 132 seconds, respectively.Reducing these parameters increased the optimal value of the solutions.We conclude that the selected parameters values guarantee (at least for these instances) a reasonable number of generations before reaching the stop criterion and sufcient diversity in the population.10 Journal of Advanced Transportation 5.3.Performance of the MA.Tis frst analysis evaluates the performance of the MA by comparing its solutions with the optimal solutions obtained with the exact model of a similar problem solved with CPLEX.In this model, buses depart at the beginning of the operation period for all lines, and no exact arrival times of passengers at the bus stops are considered.Instead, all passengers are assumed to be present at the bus stop at the beginning of each operation period.Terefore, in all instances considered in this subsection, all preferred departure times are set equal to the beginning of the operation period.Tis also implies that all passengers are served in the operation period for which they submitted a request, and the preferred departure time is always zero.Due to this simplifcation, the model can be solved to optimality for the size of instances considered.Te model is presented and discussed in Appendix A (available here).For this analysis, data for twenty operation horizons were randomly generated for each percentage of empty stops, and with fve operation periods each.Terefore, 300 operation periods are optimized both with the mathematical model in CPLEX and using the MA.Both waiting and invehicle times of passengers are evaluated, considering the standard routes for the TFS, and the optimal routes obtained by CPLEX, and the routes obtained with the MA optimization for the DRFS.Te routes for the demand illustrated in Table 3 are shown in Figure 8 as an example for the three solutions (TFS and optimal solution and MA solution for the DRFS) on all fve operation periods.Buses depart at the beginning of the operation period for all lines in the TFS and in the optimal solution for the DRFS, due to the initial assumptions.In the solution found by the MA for the DRFS, the departure times identifed by the algorithm were also equal to the beginning of the operation period for all lines.
For this analysis, data for twenty operation horizons were randomly generated for each percentage of empty stops, and with fve operation periods each.Terefore, 300 operation periods are optimized both with the mathematical model in CPLEX and using the MA.Both waiting and invehicle times of passengers are evaluated, considering the standard routes for the TFS, and the optimal routes obtained by CPLEX, and the routes obtained with the MA optimization for the DRFS.Te routes for the demand illustrated in Table 3 are shown in Figure 8 as an example for the three solutions (TFS and optimal solution and MA solution for the DRFS) on all fve operation periods.Buses depart at the beginning of the operation period for all lines in the TFS and in the optimal solution for the DRFS, due to the initial Op.Period 1st 0 2 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 2 0 0 0 0 0 1 0 11 16 64.002nd 1 2 0 1 0 2 0 1 3 2 0 1 1 0 0 2 2 1 1 0 0 0 1 0 2 23 10 40.00 3rd 0 0 3 0 2 0 0 2 1 0 0 2 2 0 2 0 2 0 2 0 2 0 0 0 0 20 15 60.00 4th 0 0 2 0 1 0 0 2 1 1 0 2 2 2 1 0 2 3 1 0 2 3 0 0 1 26 10 40.00 5th 0 0 1 0 1 1 1 1 0 1 1 0 0 0 3 1 0 0 0 6 0 1 0 1  assumptions.In the solution found by the MA for the DRFS, the departure times identifed by the algorithm were also equal to the beginning of the operation period for all lines.As mentioned before, routes are identical for all operation periods in the TFS.Six routes in the optimal DRFS solution are the same as in the TFS.It happens in the frst three operation periods for the red line and in the frst, second, and ffth period for the green line.Eleven out of ffteen for the lines in the DRFS obtained with the MA are identical to the optimal DRFS solution.All routes obtained with the MA solution were identical to the optimal solution for the frst and ffth operation periods.Tere are slight variations in the routes obtained for the second and the fourth period.Te average passenger travel time for this instance is presented in Table 5 for each operation period.For the TFS, the average passenger travel time was 31.5 min.For the DRFS, this value was 10.8% lower with the optimal solution, representing a reduction of 3.4 minutes.For the solution obtained with the MA, the average passenger travel time was 0.1 minutes longer than the optimal solution.
Te optimal travel time for passengers was achieved in four out of fve optimized periods with the MA.As mentioned before, the MA found identical routes as in the optimal solution for the frst and the last operation periods, so both simulations present the same results.For the second and the fourth operation periods, even though routes were not the same for all the lines as in the optimal solution, the obtained results with the MA were the same as the optimal result.It means that diferent routes serve passengers with the same efciency as in the optimal solutions.Passengers in the third period had an average travel time of 29 minutes with the solution of the MA, a value 2.5% above the optimal solution.
Tis same analysis was repeated for all 60 instances, comparing the average passenger travel time for the TFS, and the optimal and the MA solution for the DRFS (Table 6).In the optimal solutions, the average passenger travel time was 8.6% shorter than the TFS solution, while the MA could improve the passenger travel time by 8.0%.Moreover, the results of the MA were the same as in the optimal solution in 69.3% of the operation periods.Increasing the percentage of empty stops does not make any diference in passenger travel time on TFS because routes for the lines are identical for the whole operation period.In the optimal and the MA solutions for the DRFS, however, increasing the percentage of empty bus stops in the instances allows the possibility to skip empty bus stops and serve busy bus stops faster.With 30% empty stops, passenger travel time was 1.3 minutes shorter in the optimal solutions than in the TFS.With 70% of empty stops, this value was 4.5 minutes shorter.On average, passengers could arrive at the station 2.7 min earlier if the routes of the TFS could be adapted and changed dynamically.Comparing computation times for these simplifed instances, the mathematical models took 6.5 seconds on average to identify the optimal solution for each period of operation, and the MA took 52 seconds on average.
In the previous analysis, passengers requested their trips to start exactly at the beginning of the operation period, in order to generate a simplifed problem, allowing the comparison with results obtained by solving the mathematical model.In this second part of the analysis, the demand distribution over the diferent stops is the same as before for the 60 instances, but the preferred departure time can take any value within the operation period.Tis means that some preferred departure times occur after the bus has already served the respective bus stop, and these passengers are left to be served in the following operation period.Tis results in a variation in the average passenger waiting time, even for the TFS.For the demand presented in Table 3, the MA solution for the third operation period is illustrated in Figure 9, as an example.For this solution, the bus in the blue line departs two minutes after the operation period begins, at time 02 : 02, and arrives at the station at 02 : 35.Te bus in the green line departs at 02 : 17, and arrives at the station at 02 : 55.Te red line bus departs at 02 : 19, arriving at 02 : 53, 7 minutes before 03 : 00, the end of the period.Buses pick up passengers in the bus stops marked in the fgure.Results for this instance and routes for the fve operation periods in the MA solution are presented in Appendix B, together with the full table with the scheduled departure time for each passenger for this operation period.

Results of the MA for General Instances.
Te results for the MA are compared to the TFS for all the 60 instances, grouped by the percentage of empty stops, and presented in Table 7. Tis table shows the average passenger waiting, invehicle, and total travel time for the TFS and the DRFS with the MA solution.On average, the time passengers spend on the bus in the optimal routes of the DRFS is slightly increased compared to the time passengers took while traveling using standard routes in the TFS.However, this value decreases the more empty stops are present in the instances.It means that detours included in the lines' routes to serve passengers increase the average in-vehicle time, but skipping empty bus stops can reduce it.On the other hand, the average passenger waiting time on the DRFS reduced by 48.1% compared to the TFS, with an overall improvement of 30.6% on passenger travel time.However, since the memetic algorithm's objective is to improve passenger travel times, the routes taken by buses while not carrying passengers are not always the shortest route possible.Tis does not imply that these solutions can be further improved, because bus driving time is not part of the objective function and the buses are optimally serving passengers.Te average calculation time to achieve the termination criteria for the MA was 56 seconds per operation period.

Results of the MA for
Varying Numbers of Lines.Te previous analyses consisted of a network with three lines.In this section, the same network with the same demand is now served by a system with two and four lines instead of three, in order to analyze the impact of that parameter.To make a fair comparison with the TFS, the lines of the traditional feeder system are optimized using the MA, considering the total demand for entire operation horizon both for two and four lines.Te standard lines for the TFS services for two and four lines are presented in Appendix C.

Journal of Advanced Transportation
Te average lengths of the TFS lines with two, three, and four lines are 50.5, 31.2, and 29.3 minutes.Te reason why the average length of the lines are roughly the same for three and four lines is that the area considered can be covered easily with only three lines and therefore all three (or four) lines have a length close to the minimal travel time between the terminal and the hub station.With only two lines, longer lines are to cover the area.Terefore passengers have to wait longer for their buses and spend more time inside the vehicles before reaching the hub station.Tese results are presented in the frst columns of Table 8.
When there are four buses in the semifexible DRFS, the optimization adjusts better to the passengers' expectations.On average, the passenger travel time in the DRFS is 37.6% shorter compared to the TFS, while this was 30.6% for three lines and 29.8% for two lines on the DRFS.Te passenger waiting time reduced on average with 8.9, 14.3, and 16.8 minutes for two, three and four lines.On the other hand, it is possible to identify a reduction in the in-vehicle times of passengers for two lines only (7.9 minutes).For three and four lines, the passenger in-vehicle times increased   Te average computation time per operation period was 53 seconds for the network with 4 lines, while in the network with two lines, the computation time was 93 seconds.Longer routes also mean a search space for the MA, refected in the number of routes in the pool of alternative routes.5.6.Instances with Unrestricted Detours.So far, the possible detours that a bus could make were restricted to at most two bus stops from the original route.In this analysis, the same instances are considered, but buses are allowed to take any possible route in the network.Tis will allow to further analyze the performance of the MA and to evaluate the opportunity of allowing more fexibility in the routes.Still, no cycles are allowed, and the route length is limited by the duration of the operation period.
Te number of possible routes generated for the pool of routes increases from 192 for the red line, 60 for the green, and 225 for the blue line with at most two bus stops outside the original route (Table 1), to 2,149 routes with unrestricted detours for each line.All possible combinations from the terminal to the station are considered.Terefore, the search space increases from 2.6 million possible combinations to 10 billion.Beyond that, it is still necessary to identify the optimal departure time for the buses considering preferred departure time for the requests.
Results for the same 60 instances as in the analysis in Section 5.4 are presented in Table 9, comparing the solutions obtained with the MA for the limited and unrestricted detours.Beyond the initial reduction in travel time observed for the DRFS in comparison to the TFS, the value was further reduced by 1.7% with unrestricted detours.Te average passenger waiting time reduced from 15.4 min with limited detours to 13.7 min, 11% shorter.Te in-vehicle time was a bit longer, on average 16.6 min per passenger compared to 17.9 min with the limited detours (7.8% longer).An operation period example for this unrestricted detour analysis is presented in Appendix D. As before, increasing the percentage of empty bus stops in the simulation reduced the in-vehicle time as well, and this gain is refected in the average passenger total travel time.From 32.1 min with limited detours, the average travel time obtained with the MA solution with unrestricted detours was reduced to 31.5 min.Te average computation time for the optimizations was 407 seconds.

Large Network Instances.
One last analysis considers a larger network with 70 bus stops and a centralized hub, and 300 requests in fve operation periods.Te large network and the standard lines of the TFS are presented in Appendix C. Here, again three lines are considered to serve the bus stops, with fxed terminal on bus stops 1, 63, and 70.In order to identify the standard routes for the TFS, the MA was used with the total demand for the entire operation horizon and unrestricted detours for the lines were allowed.Tis procedure does not guarantee the optimal routes for the lines as in the previous network but leads to a good solution given the complexity of the network.Since routes are now much longer and the same numbers of buses are available, the operation period considers a frequency of one bus every two hours.Terefore, for the DRFS, the routes were limited to not exceed 120 minutes.All the other rules were kept as initially, considering the possibility of deviating from the standard route to include at most two extra bus stops or to skip stops.Fifteen instances were optimized with this network, fve for each group according to the percentage of empty stops (30%, 50%, and 70%).As before, the variation between instances occurs on the randomization of the preferred departure time of passengers and the operation period on which the trip is requested.Results are presented in Table 10.
Since the operation period is two hours and requests arrive randomly in the instances, the average passenger waiting time in the TFS is 60.1 min.For this confguration of the network and instances, the MA could be able to reduce this waiting time by 21% to 47.2 min.Te in-vehicle time was reduced by 13%, from 41.0 min on the TFS to 35.5 min on the DRFS.Te average computation time for this network was 459 seconds.Te gain on this network was smaller than Journal of Advanced Transportation the previous network, where lines could better "assist" other lines by taking over bus stops.In this larger network, stop skipping is not always used due to the unavailability of other lines to take passengers from the skipped stops.Still, signifcant improvements of on average 18.2% were obtained.

Conclusion and Future Research
Tis paper presents a semifexible demand-responsive feeder system (DRFS) as a substitute to a traditional feeder system in suburban area.Te semifexible runs a service similar to a traditional feeder system (TFS), but with adjustable routes and timetables according to passengers' requests.Its objective is to optimize passenger travel time.
From the operators' perspective, the proposed system allows to ofer a more fexible system to the passengers using the same resources.It does require a reservation system where passengers can make requests.
In order to optimize the operations of the DRFS, a memetic algorithm (MA) is developed, merging an evolutionary metaheuristic with local search operations.Te results of the MA were found to have a travel time of 0.1 minute longer than the optimal solution.Moreover, the results of the MA were the same as in the optimal solution for seven out of ten periods.
When comparing the performance of the DRFS, optimized with the MA, to the TFS for a network composed of 25 bus stops and three lines, the DRFS reduced the average passenger waiting time by 48.8% with the limited detour policy, and 53.5% with the unrestricted detour policy.However, the computation time increased from 56 to 459 seconds to optimize one operation period.Tese results prove that a semifexible DRFS allowing limited detours can provide a good solution in a faster time than a fully fexible DRFS.Despite somewhat long computation times, typical for evolutionary algorithms that perform a random exploration of the search space, these results also confrm the good performance of the MA.
When evaluating the same instances with the DRFS obtained by the MA, and comparing it with the results of the same instances in the TFS, the average passenger waiting time reduced with 31.4%, 48.1%, and 57.7% for the same network with two, three, and four lines, respectively.Despite slightly increasing the passengers' in-vehicle time with 0.6%, 0.6%, and 8.4%, respectively, the total travel time for the DRFS is always smaller than in the TFS.Tis reduction varies between 6% and 73%.
Another characteristic of of-peak operations, which are the focus of this DRFS, is that there are a lot of empty bus stops along the routes.Tis feature is explored in these experiments by categorizing the instances according to the percentage of empty stops during the operation horizon.Increasing the percentage of empty stops in an instance, from 30% over 50% to 70% increased the benefts of the DRFS.For the same network, the average passenger travel time reduced with 28.2%, 29.6%, and 34.0% for these clusters, respectively.In the network with 70 bus stops and three lines, the average passenger travel time reduced with 15.6%, 17.0%, and 22.0%, respectively.Tese results confrm the efciency of the DRFS in low-demand of-peak operations.Te overall gains, however, are dependent on the network confguration and the line characteristics, as well as the potential for shortcuts.
Te results presented in this paper demonstrated the benefts of operating a DRFS to substitute a TFS.As such, it could be used to improve efciency of transit system in suburban areas.However, the presented DRFS could be further improved.For instance, by allowing buses to arrive at the station or depart from the terminal outside the operation period and by cancelling services without passengers, the frst situation can also be observed in the results in Appendix B, where requests arriving late in the period are served during the following period only.Allowing buses to arrive at the hub station after the operation period has fnished could improve the operation and increase the responsiveness of the system.However, it also means these buses might not be available on time for the next operation period.Tis future implementation, however, would change the characteristic of the system from static to more dynamic, with a rolling horizon operation period and the possibility to reoptimize the system when new requests arrive and some services are already in operation.Tis approach is considered as future research.
Also, as mentioned in the results in Section 5.4, the bus routes are not optimized when the buses are not carrying passengers, which might lead to unnecessary detours.Terefore, an improvement could be to modify the objective function value to remove these needless detours.It means optimizing the operation from the passengers' perspective and from the operator's perspective, aiming to ofer an efcient operation in terms of vehicle kilometers as well.
Another limitation of the current DRFS is that it considers only the trips towards the hub station.Transfer times for passengers at the hub station are not considered, assuming a high frequency service from this point.In practice, hub stations concentrate several diferent line services, and transfer time coordination a crucial role to improve passengers' experience using public transport.Synchronizing these services at the station, as covered by Barabino et al. [26], and considering the transfer time at the station, as well as implementing a bi-direction service could improve the service ofered by the semifexible DRFS and increase the success of such a system.

Figure 1 :
Figure 1: Example of a line with two routes for the DRFS.

Figure 3 :
Figure 3: Blue line for a TFS operating a grid network and allowed detours and stop skipping.

Figure 2 :
Figure 2: Red line for a TFS operating a grid network and allowed detours and stop skipping.

ALGORITHM 3 :Figure 5 :
Figure 5: Example of a repair operation for a small grid network to include a request in bus stop 4 in the red or the blue line: (a) the current red line with passengers at stops 8 and 1; (b) the current blue line with passengers at stops 8 and 2; (c) fnal solution adapting the blue line.

Stop 2 ,Figure 6 :Figure 7 :
Figure 6: Illustration of improvement operation for a request and the improved line, advancing the departure time of the bus 10 minutes to reduce the waiting time of this passenger.

Figure 8 :
Figure 8: Routes for total operation in the TFS (1 st column), the optimal solution obtained with the mathematical model for the DRFS (2 nd column), and the solution obtained for the DRFS with the MA (3 rd column) for all the fve operation periods of one instance.

Figure 9 :
Figure 9: Example of MA solution for the third operation period with 50% of empty stops.
For each line in set of line: Initialize pool of possible routes for line Append requests of the current operation period in the list of requests Append rejected requests from previous periods in the list of requests YesNo Figure 4: Flowchart of the memetic algorithm.Start: For request in list of requests: If preferred departure time at request stop ≤ bus arrival at request stop: p p � time bus arrive at bus stop p d � time bus arrive at station p tt � p d − p r Solution ftness + � p tt Else: p pen � 2 * p op − p r Solution ftness + � p pen Choose random parent (P 1 ) from the list of best solutions If random: Choose random parent (P 1 ) from previous generation Choose second random parent (P 2 ) from previous generation Create empty ofspring -> O 1 , O 2 For line in set of lines: Random selection order: true or false If true: O 1 receives line confguration from P 1 O 2 receives line confguration from P 2 If false: O 1 receives line confguration from P 2 O 2 receives line confguration from P 1 ReturnO 1 , O 2

Table 1 :
Network characteristics: length of lines for the standard route in the TFS; number of routes in the pool of routes for each line in the operation period.

Table 2 :
Expected demand for each bus stop during the entire operation horizon.

Table 3 :
Example of demand distribution in the operation horizon, corresponding to fve operation periods.

Table 4 :
Sensitivity analysis of the MA for a population of 50, 100, or 200 solutions and the stop condition without improvements of the best solution for 20, 50, and 100 iterations.Results show the passenger travel time (t.t.) and the computation time (CPU).

Table 5 :
Average p tt for each period of the example simulated.

Table 6 :
Average passenger travel time in the TFS, with optimal routes and the MA.Improvement of optimal and MA solution compared to TFS, classifed according the percentage of empty bus stops in the instance.

Table 7 :
Results of average passenger waiting, in-vehicle, and total travel times for the TFS and the MA grouped by percentage of empty stops in the instances.It means that more options for stop skipping to shorten the routes are available with two lines.With three or four lines, detours optimize the passenger waiting time, but the average length of routes are roughly the same.

Table 8 :
Average passenger waiting, in-vehicle, and travel time for 2 or 4 lines.

Table 9 :
Average passenger waiting, in-vehicle, and travel time for 60 instances comparing limited detour policy and unrestricted movements for the lines.2stopsdetourUnrestricted Improvement p wt p vt p tt p wt p vt p tt p tt (%)

Table 10 :
Average passenger waiting, in-vehicle, and travel times for TFS, and the solution of the MA for the DRFS considering the large network.